Method of discriminating shape of free-form curved surface

ABSTRACT

A principle curvature of a target curved surface S′ and a principle curvature of a corresponding position of a reference surface S are obtained and each part is displayed by being classified into (a) a case where two principle curvatures increase, (b) a case where two principle curvatures decrease, and (c) a case where one of the principle curvatures increases and the other decreases from the difference between the principle curvatures. (a), (b), and (c) are determined as mountain, valley, and twist, respectively, and are displayed in different symbols or colors on an image. Consequently, a different part between two three-dimensional shapes can be accurately grasped, the cause of the occurrence of the error such as a partial curve or the like can be easily found, how much the shapes coincide with each other as a whole can be indicated by an objective numerical value, and the error can be easily determined even if the reference shape is complicated.

This application claims priority on Japanese Patent Application No.10-9139/1997, filed Apr. 25, 1997, the entire diclosure which isincorporated herein by reference.

BACKGROUND OF THE INVENTION

1. Technical Field

The present invention relates to a method of determining a shape errorof a free-form surface.

2. Background Art

With respect to formings formed by press working such as the body of anautomobile, discrepancy has been conventionally evaluated byexperiences. That is, the free-form surface has been conventionallyevaluated mainly by “visual observation”. In recent years, however,designing using a computer (CAD) has been spread and a deformation uponprocessing can be simulated. Accordingly, means for evaluating afree-form surface, that is, for objectively defining discrepancy in aforming and displaying it is desired.

FIG. 1 shows a forming sample showing discrepancy in a plate forming.FIG. 2 is a diagram showing an example of a conventional method ofevaluating a shape error in a free-form surface. The example relates toa result of numerical simulation using simulation software (ITAS-3D)reported in “Simulation of 3-D sheet bending process” (Takizawa et al.,1991, VD1 BERICHTE NR. 894), “Some advances in FEM simulation of sheetmetal forming processes using shell elements” (Kawka et al., 1995,Simulation of Materials Processing, Shen & Dawson (eds.), Balkema,Rotteerdam, pp. 735-740), and the like.

In FIG. 2, a white part shows a shape as a reference (for example, theshape of a die) and a mesh part indicates a shape obtained by a formingsimulation. The reference shape and the simulation shape are displayedat the same position and only a part positioning on the front side isdisplayed. Consequently, a shape error between the simulation shape andthe reference shape can be roughly determined from the displayed whiteand mesh parts. The method has, however, the following problems.

(1) It is necessary to determine a reference position and make thereference shape and the simulation shape accurately coincide with eachother at the reference position. The result is largely influencedaccording to the way the reference position is determined.

(2) Since the position of the other part is largely displaced due to apartial bending, it is difficult to find the cause of occurrence of anerror.

(3) How much the shapes coincide with each other as a whole cannot beshown by an objective numerical value.

FIG. 3 shows CMM data (about 40,300 points) measured by using a threecoordinate measuring machine “Mitsutoyo Super BHN 506”. FIG. 4 shows thetop view of FIG. 3 (about 8,000 points). As shown in these figures, theshape of a forming item actually formed by using a die can be displayedas images as shown in FIGS. 3 and 4 by measuring the forming item by thethree coordinate measuring machine. From the images, shape errors suchas projected and recessed parts and a twisted part can be roughlydetermined from the views. The method, however, also has theabove-mentioned problems (2) and (3) in the numerical simulation and hasa problem that (4) when the reference shape is not flat but has acomplicated curve, the difference from the result of the threecoordinate measurement can be hardly determined.

As mentioned above, methods of experiment, measurement, and display ofresult for evaluating the shape error have not been systematized yet.There has not been a simple and clear definition as an index of aforming discrepancy and, further, an evaluation method which can berepeatedly performed has not been existed conventionally.

SUMMARY OF THE INVENTION

The present invention is made in order to solve the problems. That is, aprinciple object of the invention is to provide a method of determininga shape error of a free-form surface which can accurately grasp adifferent part between two three-dimensional shapes of an actual formingshape and a simulation shape by a computer simulation, a reference shapeby CAD, or the like. It is another object of the invention to provide amethod of determining a shape error of a free-form surface which can beapplied without making the reference positions accurately coincide witheach other, find the cause of occurrence of an error such as a partialbending, show how much the shapes coincide with each other as a whole byan objective numerical value, and easily determine the error even if thereference shape is complicated.

The inventors of the present invention have invented “extended Gaussiancurvature” as an evaluation model which does not depend on thecoordinate system. According to the invention, a local shape error of afree-form surface is classified into three types (mountain, valley, andtwist) by comparing an actual curved surface with, for example, a CADcurved surface as a reference. A method of calculating the ratio of thesame labels by using the image processing technique has also beeninvented. The invention is based on the novel ideas.

According to the invention, there is provided a method of determining ashape error of a free-form surface by obtaining a principle curvature ofa target curved surface S′ and a principle curvature of a correspondingposition of a reference surface S; and displaying each part byclassifying it from the difference between the principle curvatures into(a) a case where the two principle curvatures increase, (b) a case wherethe two principle curvatures decrease, and (c) a case where one of theprinciple curvatures increases and the other decreases.

That is, according to a preferred method of the invention, Δκ₁=κ₁′−κ₁,Δκ₂=κ₂′−κ₂ are obtained from the principle curvature (κ₁′, κ₂′) of thetarget curved face S′ and the principle curvature (κ₁, κ₂) of thereference curved face S. (1) When Δκ₁≧0 and Δκ₂≧0, (a) it is determinedthat the two curvatures increase. (2) When Δκ₁≦0 and Δκ₂≦0, (b) it isdetermined that two curvatures decrease. (3) When Δκ₁·Δκ₂<0, (c) it isdetermined that one of the curvatures increases and the other decreases.Preferably, (a), (b), and (c) are determined as mountain, valley, andtwist, respectively, and are displayed in different symbols or colors onan image. Further, it is preferable that the ratio of the same labels iscalculated from the labels (a), (b), and (c) and is used as acoincidence ratio.

The Gaussian curvature K is a product κ₁κ₂ of the principle curvaturesκ₁ and κ₂ of three-dimensional surfaces. (1) When K>0, it is known thatthe shape is elliptic. (2) When K=0, it is known that the shape isparabolic. (3) When K<0, it is known that the shape is hyperbolic.

The invention relates to an extended Gaussian curvature. That is,according to the method of the invention, the principle curvature of atarget curved surface S′ including an error and the principle curvatureof a corresponding position of a reference curved surface S are obtainedand each part is classified into (a), (b), and (c) from the differencebetween the principle curvatures, namely, the parts can be displayedwhile being classified into the case where two principle curvaturesincrease, the case where two principle curvatures decrease, and the casewhere one of the principle curvatures increases and the other decreases.Thus, the different part between two three-dimensional shapes can beaccurately grasped.

According to the method, the shape error can be determined by obtainingthe principle curvatures of corresponding positions. Consequently, theinvention can be applied without making the reference positions of twothree-dimensional shapes accurately coincide with each other and thecause of occurrence of an error such as a partial bending can be found.

Further, by calculating the ratio of the same labels from the labels(a), (b), and (c) and using it as a coincidence ratio, how much theshapes coincide with each other as a whole can be grasped by anobjective numerical value and an error can be easily determined even ifthe reference shape is complicated.

The other objects and advantageous features of the invention will bemade clear from the following description with reference to the attacheddrawings.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a halftone image displayed on a display of a forming sampleshowing a discrepancy of a plate forming;

FIG. 2 is a diagram showing an example of a conventional method ofevaluating an error in a free-form surface using simulation;

FIG. 3 is a diagram of measurement by a three coordinate measuringmachine;

FIG. 4 is the top view of FIG. 3;

FIG. 5 is a halftone image displayed on a display showing a perspectiveview of an impeller;

FIG. 6 is a diagram showing that a twist deformation according to themethod of the invention is performed to an originally twisted shape;

FIG. 7 is a diagram showing that a twist deformation according to themethod of the invention is performed to an originally twisted shape;

FIG. 8 shows an example of applying the method of the invention on thebasis of measurement data;

FIG. 9 shows an example of applying the method of the invention on thebasis of the result of numerical simulation;

FIG. 10 shows another example of applying the method of the invention onthe basis of measurement data; and

FIG. 11 shows another example of applying the method of the invention onthe basis of the result of numerical simulation.

FIG. 12 outlines the steps of the method in accordance with the presentinvention.

DESCRIPTION OF THE PREFERRED EMBODIMENTS

The principle of the method of the invention will be first described.

Free-form surface S=S(u, v) is expressed by parameters u and v.Expression 1 is a relational expression in differential geometry. Therelation is disclosed in, for example, “Curves and Surfaces for ComputerAided Geometric Design” (Farin. G. 1988, A Practical Guide, AcademicPress). $\begin{matrix}\begin{matrix}{{E = \quad {S_{u}S_{u}}},{F = {S_{u}S_{v}}},{G = {S_{v}S_{v}}},{L = {nS}_{uu}}} \\{{M = \quad {nS}_{uv}},{N = {nS}_{vv}},{n = \frac{S_{u} \times S_{v}}{{}S_{u} \times S_{v}{}}}}\end{matrix} & \text{[Expression~~1]}\end{matrix}$

When it is assumed that λ=dv/du, a normal curved surface κ at anarbitrary point S (u, v) is expressed as Expression 2 in accordance withExpression 1. $\begin{matrix}{{\kappa (\lambda)} = \frac{L + {2\quad M\quad \lambda} + {N\quad \lambda^{2}}}{E + {2\quad F\quad \lambda} + {G\quad \lambda^{2}}}} & \text{[Expression~~2]}\end{matrix}$

The principle curvatures are κ₁, κ₂ which are obtained by solving thefollowing Expression 3 and the Gaussian curvature K is defined as κ₁κ₂.That is the definition of the conventional Gaussian curvature.$\begin{matrix}{{\kappa^{2} - {\frac{{NE} - {2{MF}} + {LG}}{{EG} - F^{2}}\quad \kappa} + \frac{{LN} - M^{2}}{{EG} - F^{2}}} = 0} & \text{[Expression~~3]}\end{matrix}$

The definition of an extended Gaussian curvature (Λ) invented by theinventors of the present invention can be expressed by Expression 4.$\begin{matrix}{{{{{{If}\quad ( {\kappa_{1}^{\prime} - \kappa_{1}} )\quad ( {\kappa_{2}^{\prime} - \kappa_{2}} )} \geq {0\quad {then}\quad \Lambda}} = {{{sgn}( {\kappa_{1}^{\prime} - \kappa_{1}} )}( {\kappa_{1}^{\prime} - \kappa_{1}} )( {\kappa_{2}^{\prime} - \kappa_{2}} )}}\quad {{else}\quad \Lambda} = {{abs}\quad ( {( {\kappa_{1}^{\prime} - \kappa_{1}} )\quad ( {\kappa_{2}^{\prime} - \kappa_{2}} )} )}}\quad {{\quad {sgn}\quad (a)} = \{ {\begin{matrix}{1( {a \geq 0} )} \\{{- 1}( {a < 0} )}\end{matrix},{{{abs}\quad (a)} = \{ \begin{matrix}{a( {a \geq 0} )} \\{- {a( {a < 0} )}}\end{matrix} }} }} & \text{[Expression~~~4]}\end{matrix}$

That is, according to the method of the invention, first, Δκ₁=κ₁′−κ₁ andΔκ₂=κ₂′−κ₂ are obtained from the principle curvature (κ₁′, κ₂′) of thetarget curved surface S′ and the principle curvature (κ₁, κ₂) of thereference curved surface S. (1) When Δκ₁≧0 and Δκ₂≧0, (a) it isdetermined that the two curvatures increase. (2) When Δκ₁≦0 and Δκ₂≦0,(b) it is determined that two curvatures decrease. (3) When Δκ₁·Δκ₂<0,(c) it is determined that one of the curvatures increases and the otherdecreases. (a), (b), and (c) are discriminated as mountain, valley, andtwist, respectively, and are displayed by different symbols or colors onan image.

In other words,

(1) when (κ₁′−κ₁)(κ₂′−κ₂)≧0 and (κ₁′−κ₁)≧0, a label “mountain” isappended to Λ.

(2) When (κ₁′−κ₁)(κ₂′−κ₂)≧0 and (κ₁′−κ₁)<0, a label “valley” is appendedto Λ.

(3) When (κ₁′−κ₁)(κ₂′−κ₂)<0, a label “twist” is appended to Λ.

[Embodiments]

Embodiments in which the method of the invention is applied will bedescribed hereinbelow with reference to the drawings.

(Embodiment 1)

FIG. 5 is a perspective view of an impeller. FIGS. 6 and 7 show theresults of application of the method of the invention. FIG. 6 shows acase where a blade is bent and FIG. 7 shows a case where the blade istwisted by applying forces shown by the arrows.

In FIGS. 6 and 7, the labels “mountain”, “valley”, and “twist” accordingto the invention are indicated by symbols “+”, “−”, and “% ”,respectively. In an actual image display, it is preferable to show the“mountain”, “valley”, and “twist” in colors such as brown, blue, andred, respectively. By displaying the determination of the shape error, adifferent part from the original three-dimensional curved face can begrasped easily and accurately by the different labels or differentcolors.

According to the method of the invention, the ratio of the same labelsis calculated from the labels (a), (b), and (c) and is used as acoincidence ratio. That is, the labels of “mountain”, “valley”, and“twist” are mapped on pixels of a parameter plane ([0, 1]×[0, 1]) of u,v and the parameter plane is divided into lattice at a proper pitch (d).By applying a coincidence ratio (ψ) of Expression 5, how much the shapescoincide with each other can be shown by an objective numerical value.The coincidence ratio (ψ) indicates the relevance factor of the kind ofthe label. Since there is conventionally no index of discrepancy in aforming item, the coincidence ratio (ψ) can be used as a simple andclear index. It can be further developed and can be further finelydivided in accordance with the magnitude of the extended Gaussiancurvature (Λ). $\begin{matrix}{\psi = {\frac{( {{number\_ of}{\_ the}{\_ same}{\_ label}} )}{d^{2}} \times 100\quad (\%)}} & \text{[Expression~~~5]}\end{matrix}$

(Embodiment 2)

FIG. 8 shows an application example of the method of invention on thebasis of the measurement data of FIG. 4. FIG. 9 shows an applicationexample of the method of the invention on the basis of the numericalsimulation result of FIG. 2. That is, FIGS. 8 and 9 show CMM data andFEM data, respectively. The data has been subjected to surface fittingof a solid modeler “DESIGNBASE” (manufactured by Richo) with anallowance of 0.002 mm from the original point. In this case, thecoincidence ratio (ψ) is 50.23%.

Although the labels “mountain”, “valley”, and “twist” in the inventionare shown by symbols of +, −, and %, respectively, also in FIGS. 8 and9, it is preferable that the mountain, valley, and twist are shown bycolors such as brown, blue, and red, respectively, in an actual imagedisplay.

It will be understood from FIGS. 8 and 9 that the difference between theactual forming shape (FIG. 8) or the simulation shape (FIG. 9) obtainedby computer simulation and the reference shape, that is, a differentpart between the two three-dimensional shapes can be accurately grasped.In this regards, the method of the invention is more excellent than theconventional method shown in FIGS. 2 to 4 by far.

Since the shape error can be determined by obtaining the principlecurvatures of corresponding positions, the method can be applied withoutaccurately coinciding the reference positions of two three-dimensionalshapes with each other, and the cause of the shape error such as apartial curve can be found.

Further, by calculating the ratio of the same labels from the labels(a), (b), and (c) and using it as a coincidence ratio, how much theshapes coincide with each other as a whole can be objectively shown by anumerical value and an error can be easily discriminated even when thereference shape is complicated.

(Embodiment 3)

FIGS. 10 and 11 show comparison examples with respect to the “side face”of FIG. 1. FIG. 10 shows an application example of the method of theinvention on the basis of the measurement data. FIG. 11 shows anapplication example of the method of the invention on the basis of theresult of the numerical simulation. The coincidence ratio (ψ) in thiscase is 52.47%.

POSSIBILITY OF INDUSTRIAL UTILIZATION

As mentioned above, the invention provides a simple and general methodof defining a local shape error in a free-form surface. According to themethod, the difference between the principle curvature of a curvedsurface including an error and that of a reference curved surface isused. The reference curved surface is usually expressed by CAD data. Thecurved surface including an error is obtained by approximating a groupof discrete points such as measurement points and node points innumerical simulation. The principle curvature is used to evaluate acurved surface from the viewpoint of design, to form a curved surface,and the like, but is conventionally not used numerically to formulate orcompare a shape error. The inventors of the present invention formulateda local shape error and performed FEM simulation and CMM data comparisonby using the CAD data as a reference.

According to the method of the invention, “extended Gaussian curvature”is newly defined and a local shape error in a free-form surface isclassified into three types (mountain, valley, and twist) by comparingthe free-from surface with a CAD curved surface as a reference. Thecoincidence ratio calculating method using the image process techniqueis also proposed. With respect to the example of evaluating the formingdiscrepancy, the method of the invention was proved to be effective andmighty from the comparison of the deviation pattern of the actualmeasurement data and the numerical simulation data from the CAD datawhich is referred to.

By using the method of the invention, the discrepancy of the formingitem which is formed by a press work such as the body of an automobileor the like can be evaluated objectively and numerically from, forexample, the actual measurement data or the accuracy of the numericalsimulation can be evaluated objectively and numerically. Consequently,the method of the invention can be widely industrially used with a threecoordinate measuring machine, a CAD system, a CAM system, or asimulation system in the wide-ranged forming processing field or theshape measuring and evaluating field.

As mentioned above, according to the method of determining the shapeerror in the free-form surface of the invention, there are excellenteffects such that the different part between two three-dimensionalshapes can be accurately grasped, the shape error can be determined byobtaining the principle curvatures of the corresponding positions, thecause of the occurrence of an error such as a partial curve can befound, how much the shapes coincide with each other as a whole can beshown by an objective numerical value as a ratio, and the error can beeasily determined even in a case where the reference shape iscomplicated.

Although the invention has been described in accordance with somepreferred embodiments, it can be understood the range of the rightincluded in the invention is not limited by the embodiments. On thecontrary, the range of the right of the invention includes all ofimprovements, modifications, and equivalents included in the scope ofthe appended claims.

What is claimed is:
 1. A method of determining a shape error in afree-form surface, comprising the steps of: obtaining a principlecurvature of a target curved surface S′ and a principle curvature of acorresponding position of a reference surface S; and displaying eachpart while classifying the principle curvatures into (a) a case wherethe two principle curvatures increase, (b) a case where the twoprinciple curvatures decrease, and (c) a case where one of the principlecurvatures increases and the other decreases from the difference betweenthe principle curvatures.
 2. The method of determining a shape error ina free-form surface according to claim 1, wherein Δκ₁=κ₁′−κ₁ andΔκ₂=κ₂′−κ₂ are obtained from the principle curvature (κ₁′, κ₂′) of thetarget curved surface S′ and the principle curvature (κ₁, κ₂) at thecorresponding position of the reference curved surface S, (1) when Δκ₁≧0and Δκ₂≧0, (a) it is determined that the two curvatures increase, (2)when Δκ₁≦0 and Δκ≦0, (b) it is determined that two curvatures decrease,and (3) when Δκ₁·Δκ₂<0, (c) it is determined that one of the principlecurvatures increases and the other decreases.
 3. The method ofdetermining a shape error in a free-form surface according to claim 1,wherein (a), (b) and (c) are determined as mountain, valley, and twist,respectively, and are displayed in different symbols or colors on animage.
 4. The method of determining a shape error in a free-form surfaceaccording to claim 1, wherein the ratio of the same labels is calculatedfrom the labels (a), (b), and (c) and is used as a coincidence ratio. 5.The method of determining a shape error in a free-form surface accordingto claim 4, wherein the absolute value of Δκ₁·Δκ₂ is calculated everylabel of (a), (b), and (c) and is quantified.
 6. The method ofdetermining a shape error in a free-form surface according to claim 2,wherein (a), (b), and (c) are determined as mountain, valley, and twist,respectively, and are displayed in different symbols or colors on animage.
 7. The method of determining a shape error in a free-form surfaceaccording to claim 2, wherein the ratio of the same labels is calculatedfrom the labels (a), (b), and (c) and is used as a coincidence ratio. 8.The method of determining a shape error in a free-form surface accordingto claim 3, wherein the ratio of the same labels is calculated from thelabels (a), (b), and (c) and is used as a coincidence ratio.
 9. Themethod of determining a shape error in a free-form surface according toclaim 3, wherein the absolute value of Δκ₁·Δκ₂ is calculated every labelof (a), (b), and (c) and is quantified.
 10. The method of determining ashape error in a free-form surface according to claim 4, wherein theabsolute value of Δκ₁·Δκ₂ is calculated every label of (a), (b), and (c)and is quantified.